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We consider ideals I of subsets of the set of natural numbers N such that for every conditionally convergent series of real numbers ∑ n∈N a n and s ∈ R, then there is a sequence of signs δ = (δ n) n∈N such that ∑ n∈N δ n a n = s and N(δ) := {n ∈ N : δ n = −1} ∈ I. We give some properties of such ideals and characterize them in terms of extendability to a… (More)